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Algebra. 226. TO FIND Two NUMBERS WHEN THEIR SUM AND

DIFFERENCE ARE GIVEN.

1. The difference of two numbers is 4 and their sum is 20. What are the numbers ? Let

x = the smaller number, then x + 4 = the larger number,

and x + x + 4 20. Transposing x + x = 20 - 4. Uniting

16. Dividing

8, the smaller number x + 4 = 12, the larger number.

2 x

X =

2. The difference of two numbers is 9 and their sum is 119. What are the numbers ?

3. The difference of two fractions is ao and their sum is . What are the fractions ?

4. The difference of two numbers is d and their sum is s. What are the numbers ?

= S

Let

X = the smaller number then x + d = the larger number,

and x + x + d Transposing x + x = s d Uniting

2 x = s

d

d Dividing

2

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Observe that any number you please may be put in the place of s; and any number less than s, in the place of d: so when the sum and the difference of two numbers are given the smaller number may be found by subtracting the difference from the sum and dividing the remainder by 2.

5. The difference of two numbers is 327, and their sum is 1159. What are the numbers ?

Algebra.

227. ANOTHER METHOD OF FINDING Two NUMBERS WHEN

THEIR SUM AND DIFFERENCE ARE GIVEN.

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1. The difference of two numbers is 17 and their sum is 69. What are the numbers ? Let

x = the larger number, then

17 = the smaller number, and x + x 17 = 69. Transposing x + x = 69 + 17. Uniting

86. Dividing

x = 43, the larger number,

26, the smaller number.

2 x

x – 17

2. The difference of two numbers is 8.4 and their sum 75.6. What are the numbers ?

3. The difference of two numbers is d and their sum is s. What are the numbers ?

Let

x = the larger number, then x d = the smaller number,

and x + x - d Transposing x + x = std Uniting

= s + d

= S

2 x

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Observe that any number you please may be put in the place of s and any number less than s in the place of d; so when the sum and difference of two numbers are given the larger may be found by adding the difference to the sum and dividing the amount by 2.

4. A horse and a harness together are worth $146, and the horse is worth $74 more than the harness. Find the value of each.

Geometry. 228. How MANY DEGREES IN EACH ANGLE OF A REGU

LAR PENTAGON ?

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1. Every regular pentagon may be divided into equal, isosceles triangles.

2. The sum of the angles of one triangle is equal to right angles ; * then the sum of the angles of 5 triangles is equal to — right angles.

3. But the sum of the central angles in figure 2, (a + b + c+d+e) is equal to – right angles ; † then the sum of all the other angles of the five triangles is equal to 10 right angles, less 4 right angles, or 6 right angles = 540°. But the angular space that measures 540°, as shown in Figure 2, is made up of 10 equal angles, so each one of the angles is 1 tenth of 540° or 54o. Two of these angles, as 1 and 2, make one of the angles of the pentagon ; therefore each angle of the pentagon measures 2 times 54° or 108°.

4. Using the protractor construct a regular pentagon as follows:

(a) Draw two lines that meet in a point, each line being 2 inches long and the angular space between them being 108°.

(b) Regarding the two lines as two sides of a regular pentagon, draw two more sides each 2 inches long and joining those already drawn at an angle of 108.

(c) Complete the figure by drawing the fifth side and prove your work by measuring the last line drawn and the other two angles of the pentagon.

* See page 59, 6 and 7. + See page 29, Art. 66.

229. MISCELLANEOUS REVIEW. Remembering that in speaking of the per cent of loss or gain, the cost is the base unless otherwise specified, tell the per cent of loss or gain in each of the following:

1. Bought for 2 and sold for 3.
2. Bought for 3 and sold for 2.
3. Bought for 4 and sold for 5.
4. Bought for 5 and sold for 4.
5. Bought for 5 and sold for 6.

6. Bought for 6 and sold for 5.
7. Bought for 8 and sold for 10; for 12.
8. Bought for 8 and sold for 14 ; for 16.
9. Bought for 8 and sold for 18; for 20.

10. Bought for 8 and sold for 4 ; for 2. 11. Mr. Parker sold goods at a profit of 25 %; the amount of his sales on a certain day was $24.60. How much was his profit?*

12. Mr. Jewell sold goods at a loss of 25 %; the amount of his sales on a certain day was $24.60. How much was his loss ? *

13. By selling a horse for $156 there was a loss to the seller of 20 %. What would have been his gain per cent if he had sold the horse for $234 ? †

14. A bill was made for wood that was supposed to be 4 feet long. It was afterwards found to be only 46 inches long. What % should be deducted from the bill?

15. The marked price on a pair of boots was 25 % above cost. If the dealer sells them for 25 % less than the marked price will he receive more or less than the cost of the boots?

16. If by selling goods at a profit of 12% a man gains $6.60, what was the cost of the goods ?

* $24.60 is what per cent of the cost ? + First find the cost of the horse.

PERCENTAGE.

230. DISCOUNTING BILLS.

Many kinds of goods are usually sold “on time;" that is, the buyer may have 30, 60, or 90 days in which to pay for them. If he pays for such goods at the time of purchase or within ten days from the time of purchase, his bill is “discounted ” from 1 % to 6 %, accord. ing to agreement; that is, a certain part of the amount of the bill is deducted from the amount.

EXAMPLE.

Mr. Smith bought of Marshall Field & Co. a bill of goods amounting to $350.20. The discount for immediate payment (“spot cash ") was 1 %. How much must he pay for the goods?

1% of $350.20 is $3.50. $350.20 – $3.50 = $346.70.

PROBLEMS.
“Figure the discounts” on the following bills: †

1. Bill of $324.37, discounted at 2%.
2. Bill of $276.45, discounted at 1 %.
3. Bill of $356.50, discounted at 3%.
4. Bill of $536.50, discounted at 6 %.

5. Bill of $561.80, discounted at 4 %. (a) Find the sum of the five bills before they are discounted.

(b) Find the sum of the discounts.
(c) Find the sum of the five bills after discounting.

+ Fractions of cents in the discount on each bill may be ignored.

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